We multiply each of these by six and we keep the same order. So that is equivalent right over there. Let's see, to go from seven to 63, you multiply by nine.
And to go from six to 54, you also multiply by nine. So once again, 63 to 54 is an equivalent ratio. And so we've already selected three, but let's just verify that this doesn't work. So to go from seven to 84, you would multiply by Let's do another example. So once again, we are asked to select three ratios that are equivalent to 16 to So pause this video and see if you can work through it.
Alright, let's look at this first one. So eight to six. So at first you might say well, gee, these numbers are smaller than 16 and Remember, you can, to get an equivalent ratio you can multiply or divide these numbers by the same number. So, to get from 16 to eight, you could do that as, well, we just divided by two. And to go from 12 to six, you also divide by two. So this actually is an equivalent ratio. I'll circle that in. What about 32 to 24? Well to go from 16 to 32, we multiply by two.
To go from 12 to 24, we also multiply by two. So this is an equivalent ratio. What about four to three? Well, to go from 16 to four, we would have to divide by four. And to go from 12 to three, we are going to divide by four as well. So we're dividing by the same thing, each of these numbers. So, this is also going to be an equivalent ratio. So we've selected our three, so we are essentially done. But, we might as well see why these don't work. Now let's think about it.
To go from 16 to 12, how do we do that? Well, to go from 16 to 12, you could divide by four and multiply by three. You would get And to go from 12 to eight, so you could divide by three and multiply by two. So you'd be multiplying or dividing by different numbers here, so this one is not equivalent. And then 24 to 16? So you're not multiplying by the same amount. So once again, not an equivalent ratio. Hence, the effective interest rate on the investment is You then rewrite the future value formula:.
On the left-hand side, the interest amount divided by the present value results in the interest rate:. Formula 9. It further adapts to any conversion between different compounding frequencies.
Step 2 : Apply Formula 9. Step 3 : Apply Formula 9. Revisiting the opening scenario, comparing the interest rates of 6. Therefore, you could convert both nominal interest rates to effective rates. The rate of 6. The better mortgage rate is 6. Use this function to solve for any of the three variables, not just the effective rate.
To use this function, enter two of the three variables by keying in each piece of data and pressing ENTER to store it. When you are ready to solve for the unknown variable, scroll to bring it up on your display and press CPT. For example, use this sequence to find the effective rate equivalent to the nominal rate of 6.
Annually compounded interest rates can be used to quickly answer a common question: "How long does it take for my money to double? Written algebraically this is. Note how close the approximations are to the actual times. If your investment earns 5.
According to the Rule of 72, approximately how long will it take your investment to double at this effective rate? Once known, apply the Rule of 72 to approximate the doubling time. To answer approximately how long it will take for the money to double, apply the Rule of You are effectively earning 5.
As you search for a car loan, all banks have quoted you monthly compounded rates which are typical for car loans , with the lowest being 8. At your last stop, the credit union agent says that by taking out a car loan with them, you would effectively be charged 8. Should you go with the bank loan or the credit union loan? Thus, steps 2 and 3 are performed in the opposite order. The offer of 8. If the lowest rate from the banks is 8. At times you must convert a nominal interest rate to another nominal interest rate that is not an effective rate.
For example, in the opening scenario of this section your mortgage rates were all quoted semi-annually except for one monthly rate. One way to compare these rates was to make them all effective rates. An alternative is to take the "oddball" rate and convert it to match the compounding of all the other rates.
This brings up the concept of equivalent interest rates, which are interest rates with different compounding that produce the same effective rate and therefore are equal to each other.
After one year, two equivalent rates have the same future value. To convert nominal interest rates you need no new formula. Instead, you make minor changes to the effective interest rate procedure and add an extra step. Follow these steps to calculate any equivalent interest rate:. Step 4 : Apply Formula 9. Once again revisiting the mortgage rates from the section opener, compare the 6.
It is arbitrary which interest rate you convert. In this case, choose to convert the 6. Step 2 : Applying Formula 9. Step 3 : Applying Formula 9. Step 6 : Applying Formula 9. Thus, 6. Pick the mortgage rate of 6. Of course, this is the same decision you reached earlier.
Step 1 : Convert the original nominal rate and compounding to an effective rate.
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